Problem: Find the distance between the point ${(3, 4)}$ and the line $\enspace {y = 7}\thinspace$. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$
Solution: First, find the equation of the perpendicular line that passes through ${(3, 4)}$ Since the slope of the blue line is $0$ , the perpendicular line will have an infinite slope and therefore will be a vertical line. The equation of the vertical line that passes through ${(3, 4)}$ is $\enspace {x = 3}\thinspace$ We can see from the graph that the two lines intersect at the point ${(3, 7)}$ . Thus, the distance we're looking for is the distance between the two red points. Since their $x$ components are the same, the distance between the two points is simply the change in $y$ $|{4} - ( {7} )| = 3$ The distance between the point ${(3, 4)}$ and the line $\enspace {y = 7}\enspace$ is $\thinspace3$.